Finite Group DatabasebetaFinite Group Databasebeta| order | 1568 |
| isoclinism | isoclinic class of #5997 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C7 |
No aliases for this group.
| derived, Sylow | C72 ⋊ C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | #5997 |
| commutator, hypocenter | C72 | C32 |
| Fitting | C72×C8 | C4 |
| socle, prime radical | C2×C72 | C16 |
| cosocle | C7⋊(C7⋊C16) | C2 |
| LCS | G ▷ C72 | |
| UCS | C1 ◁ C8 | |
32 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^5,7^2|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^3g^5,g^a=f^5g^4,f^b=f^6,g^b=g^6128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1568, 198);